Skip to main content

Computing Bisimilarity Metrics for Probabilistic Timed Automata

  • Conference paper
  • First Online:
Integrated Formal Methods (IFM 2019)

Abstract

We are interested in describing timed systems that exhibit probabilistic behaviour and in evaluating their behavioural discrepancies. To this purpose, we consider probabilistic timed automata, we introduce a concept of n-bisimilarity metric and give an algorithm to decide it.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Recall that probabilities and \(d_{{\mathsf {G}}}(z)\) are rationals.

References

  1. de Alfaro, L., Faella, M., Stoelinga, M.: Linear and branching system metrics. IEEE Trans. Softw. Eng. 35(2), 258–273 (2009)

    Article  Google Scholar 

  2. de Bakker, J.W., Huizing, C., de Roever, W.P., Rozenberg, G. (eds.): REX 1991. LNCS, vol. 600. Springer, Heidelberg (1992). https://doi.org/10.1007/BFb0031984

    Book  Google Scholar 

  3. Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126, 183–235 (1994)

    Article  MathSciNet  Google Scholar 

  4. Asarin, E., Basset, N., Degorre, A.: Distance on timed words and applications. In: Jansen, D.N., Prabhakar, P. (eds.) FORMATS 2018. LNCS, vol. 11022, pp. 199–214. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-00151-3_12

    Chapter  MATH  Google Scholar 

  5. Bacci, G., Bacci, G., Larsen, K.G., Mardare, R.: On-the-fly exact computation of bisimilarity distances. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 1–15. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36742-7_1

    Chapter  MATH  Google Scholar 

  6. Bacci, G., Bacci, G., Larsen, K.G., Mardare, R.: On-the-fly computation of bisimilarity distances. Logical Methods Comput. Sci. 13(2) (2017)

    Google Scholar 

  7. Bacci, G., Bacci, G., Larsen, K.G., Mardare, R.: Converging from branching to linear metrics on Markov Chains. Math. Struct. Comp. Sci. 29(1), 3–37 (2019)

    Article  MathSciNet  Google Scholar 

  8. Bacci, G., Bacci, G., Larsen, K.G., Mardare, R., Tang, Q., van Breugel: F.: Computing probabilistic bisimilarity distances for probabilistic automata. In: CONCUR. LIPiCS, vol. 140, pp. 9:1–9:17 (2019)

    Google Scholar 

  9. Beauquier, D.: On probabilistic timed automata. Theor. Comput. Sci. 292, 65–84 (2003)

    Article  MathSciNet  Google Scholar 

  10. Bertrand, N., et al.: Stochastic timed automata. Log. Meth. Comp. Sci. 10 (2014)

    Google Scholar 

  11. Bouyer, P.: Forward analysis of updatable timed automata. Formal Meth. Syst, Des. 24, 281–320 (2004)

    Article  Google Scholar 

  12. Bouyer, P., Dufourd, C., Fleury, E., Petit, A.: Updatable timed automata. Theoret. Comput. Sci. 321(2–3), 291–345 (2004)

    Article  MathSciNet  Google Scholar 

  13. van Breugel, F.: On behavioural pseudometrics and closure ordinals. Inf. Process. Lett. 112(19), 715–718 (2012)

    Article  MathSciNet  Google Scholar 

  14. van Breugel, F., Worrell, J.: A behavioural pseudometric for probabilistic transition systems. Theoret. Comput. Sci. 331(1), 115–142 (2005)

    Article  MathSciNet  Google Scholar 

  15. van Breugel, F., Sharma, B., Worrell, J.: Approximating a behavioural pseudometric without discount. Log. Methods Comput. Sci. 4(2) (2008)

    Google Scholar 

  16. van Breugel, F., Worrell, J.: Approximating and computing behavioural distances in probabilistic transition systems. Theor. Comput. Sci. 360(1), 373–385 (2006)

    Article  MathSciNet  Google Scholar 

  17. van Breugel, F., Worrell, J.: The complexity of computing a bisimilarity pseudometric on probabilistic automata. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds.) Horizons of the Mind. A Tribute to Prakash Panangaden. LNCS, vol. 8464, pp. 191–213. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06880-0_10

    Chapter  Google Scholar 

  18. Castiglioni, V., Gebler, D., Tini, S.: Logical characterization of bisimulation metrics. In: QAPL 2016. EPTCS, vol. 227, pp. 44–62 (2016)

    Article  MathSciNet  Google Scholar 

  19. Castiglioni, V., Tini, S.: Logical characterization of branching metrics for nondeterministic probabilistic transition systems. Inf. Comput. 268 (2019)

    Article  MathSciNet  Google Scholar 

  20. Čerāns, K.: Decidability of bisimulation equivalences for parallel timer processes. In: von Bochmann, G., Probst, D.K. (eds.) CAV 1992. LNCS, vol. 663, pp. 302–315. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56496-9_24

    Chapter  Google Scholar 

  21. Chatterjee, K., Ibsen-Jensen, R., Majumdar, R.: Edit distance for timed automata. In: HSCC14, pp. 303–312. ACM (2014)

    Google Scholar 

  22. Chatterjee, K., Prabhu, V.S.: Quantitative temporal simulation and refinement distances for timed systems. IEEE Trans. Automat. Contr. 60(9), 2291–2306 (2015)

    Article  MathSciNet  Google Scholar 

  23. Chen, D., van Breugel, F., Worrell, J.: On the complexity of computing probabilistic bisimilarity. In: Birkedal, L. (ed.) FoSSaCS 2012. LNCS, vol. 7213, pp. 437–451. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28729-9_29

    Chapter  MATH  Google Scholar 

  24. Chen, T., Han, T., Katoen, J.: Time-abstracting bisimulation for probabilistic timed automata. In: TASE08, pp. 177–184. IEEE (2008)

    Google Scholar 

  25. David, A., Larsen, K.G., Legay, A., Mikučionis, M., Poulsen, D.B.: Uppaal SMC tutorial. STTT 17(4), 397–415 (2015)

    Article  Google Scholar 

  26. Deng, Y., Chothia, T., Palamidessi, C., Pang, J.: Metrics for action-labelled quantitative transition systems. QAPL. ENTCS 153(2), 79–96 (2006)

    Google Scholar 

  27. Deng, Y., Du, W.: The kantorovich metric in computer science: a brief survey. QAPL. ENTCS 253(3), 73–82 (2009)

    Google Scholar 

  28. Desharnais, J., Gupta, J., Jagadeesan, R., Panangaden, P.: Metrics for labelled Markov processes. Theoret. Comput. Sci. 318(3), 323–354 (2004)

    Article  MathSciNet  Google Scholar 

  29. Dill, D.L.: Timing assumptions and verification of finite-state concurrent systems. In: Sifakis, J. (ed.) CAV 1989. LNCS, vol. 407, pp. 197–212. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-52148-8_17

    Chapter  Google Scholar 

  30. Ferns, N., Panangaden, P., Precup, D.: Metrics for finite markov decision processes. In: UAI, pp. 162–169. AUAI Press (2004)

    Google Scholar 

  31. Giacalone, A., Jou, C., Smolka, S.A.: Algebraic reasoning for probabilistic concurrent systems. In: IFIP TC2 PROCOMET (1990)

    Google Scholar 

  32. Henzinger, T., Nicollin, X., Sifakis, J., Yovine, S.: Symbolic model checking for real-time systems. Inf. Comp. 111(2), 193–244 (1994)

    Article  MathSciNet  Google Scholar 

  33. Henzinger, T.A., Majumdar, R., Prabhu, V.S.: Quantifying similarities between timed systems. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829, pp. 226–241. Springer, Heidelberg (2005). https://doi.org/10.1007/11603009_18

    Chapter  MATH  Google Scholar 

  34. Kwiatkowska, M., Norman, G., Segala, R., Sproston, J.: Automatic verification of real-time systems with discrete probability distributions. Theoret. Comput. Sci. 282, 101–150 (2002)

    Article  MathSciNet  Google Scholar 

  35. Kwiatkowska, M., Norman, G.: Probabilistic metric semantics for a simple language with recursion. In: Penczek, W., Szałas, A. (eds.) MFCS 1996. LNCS, vol. 1113, pp. 419–430. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61550-4_167

    Chapter  Google Scholar 

  36. Lanotte, R., Maggiolo-Schettini, A., Troina, A.: Weak bisimulation for probabilistic timed automata. Theor. Comput. Sci. 411(50), 4291–4322 (2010)

    Article  MathSciNet  Google Scholar 

  37. Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991)

    Article  MathSciNet  Google Scholar 

  38. Majumdar, R., Prabhu, V.S.: Computing the Skorokhod distance between polygonal traces. In: HSCC15, pp. 199–208. ACM (2015)

    Google Scholar 

  39. Norman, G., Parker, D., Sproston, J.: Model checking for probabilistic timed automata. Formal Methods Syst. Des. 43, 164–190 (2013)

    Article  Google Scholar 

  40. Segala, R.: Modeling and verification of randomized distributed real-time systems. Ph.D. thesis, MIT (1995)

    Google Scholar 

  41. Sen, K., Viswanathan, M., Agha, G.: Statistical model checking of black-box probabilistic systems. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 202–215. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-27813-9_16

    Chapter  Google Scholar 

  42. Sproston, J., Troina, A.: Simulation and bisimulation for probabilistic timed automata. In: Chatterjee, K., Henzinger, T.A. (eds.) FORMATS 2010. LNCS, vol. 6246, pp. 213–227. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15297-9_17

    Chapter  MATH  Google Scholar 

  43. Stoelinga, M.: Alea jacta est: verification of probabilistic, real-time and parametric systems. Ph.D. thesis, University of Nijmegen, The Netherlands (2002)

    Google Scholar 

  44. Tang, Q., van Breugel, F.: Computing probabilistic bisimilarity distances via policy iteration. In: CONCUR, pp. 22:1–22:15. LIPIcs (2016)

    Google Scholar 

  45. Tang, Q., van Breugel, F.: Algorithms to compute probabilistic bisimilarity distances for Labelled Markov Chains. In: CONCUR, pp. 27:1–27:16. LIPiCS (2017)

    Google Scholar 

  46. TaŞiran, S., Alur, R., Kurshan, R.P., Brayton, R.K.: Verifying abstractions of timed systems. In: Montanari, U., Sassone, V. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 546–562. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61604-7_75

    Chapter  Google Scholar 

  47. Yamane, S.: Probabilistic timed simulation verification and its application to stepwise refinement of real-time systems. In: Saraswat, V.A. (ed.) ASIAN 2003. LNCS, vol. 2896, pp. 276–290. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-40965-6_18

    Chapter  Google Scholar 

  48. Younes, H.L.S.: Verification and planning for stochastic processes with asynchronous events. Ph.D. thesis, Pittsburgh, PA, USA (2004)

    Google Scholar 

  49. Yovine, S.: Model checking timed automata. In: Rozenberg, G., Vaandrager, F.W. (eds.) EEF School 1996. LNCS, vol. 1494, pp. 114–152. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-65193-4_20

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Simone Tini .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lanotte, R., Tini, S. (2019). Computing Bisimilarity Metrics for Probabilistic Timed Automata. In: Ahrendt, W., Tapia Tarifa, S. (eds) Integrated Formal Methods. IFM 2019. Lecture Notes in Computer Science(), vol 11918. Springer, Cham. https://doi.org/10.1007/978-3-030-34968-4_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-34968-4_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-34967-7

  • Online ISBN: 978-3-030-34968-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics